Preservation of positivity by dynamical coarse graining

Abstract

We compare different quantum Master equations for the time evolution of the reduced density matrix. The widely applied secular approximation (rotating wave approximation) applied in combination with the Born-Markov approximation generates a Lindblad type master equation ensuring for completely positive and stable evolution and is typically well applicable for optical baths. For phonon baths however, the secular approximation is expected to be invalid. The usual Markovian master equation does not generally preserve positivity of the density matrix. As a solution we propose a coarse-graining approach with a dynamically adapted coarse graining time scale. For some simple examples we demonstrate that this preserves the accuracy of the integro-differential Born equation. For large times we analytically show that the secular approximation master equation is recovered. The method can in principle be extended to systems with a dynamically changing system Hamiltonian, which is of special interest for adiabatic quantum computation. We give some numerical examples for the spin-boson model of cases where a spin system thermalizes rapidly, and other examples where thermalization is not reached.

Systematic perturbation theory for dynamical coarse-graining

“…The conventional Born-Markov-Secular limit is obtained by the limit τ → ∞, i.e.,ρ S BMS = L ∞ ρ BMS S , whereas in the short-time limit, the exact full solution is approximated. In addition, it was found for some simple examples considered in [24] that in the weak coupling limit, the method approximated the results of the non-Markovian master equation for all times remarkably well.…”

“…Since the Liouville superoperators L τ are of Lindblad form [20] for all τ > 0, the second-order dynamical coarse-graining approach (DCG2) preserves positivity of the density matrix at all times [24]. Note that in the general case, the above solution cannot be obtained by solving a single Lindblad form master equation merely equipped with time-dependent coefficients and should therefore be regarded as truly non-Markovian [21].…”

“…This will evidently yield good results for small λ and small times, whereas we would like to have a master equation valid for small λ and also large times. In the original approach [24] it was shown as a supportive fact that for t = τ the DCG2 solution and the approximation (11) were equivalent up to O{λ 2 }. Here we will demand equivalence between the n-th order coarse graining solution and Eqn.…”

“…In complete analogy to ref [24] we obtain the Born-Markov secular approximation [4] in this limit. Unfortunately, the unconditional preservation of positivity is not preserved by higher orders within DCG (although of course, in the weak coupling limit the nice properties of DCG2 will dominate).…”

“…Recently, it been analyzed up to second order in the system-reservoir coupling constant (Born approximation) [24]. Instead of solving a single quantum master equation, the coarse-graining approach defines a continuous set of master equationṡ…”